NBER WORKING PAPER SERIES

DESIGNING EFFICIENT CONTACT TRACING THROUGH RISK-BASED QUARANTINING

Andrew PerraultMarie CharpignonJonathan GruberMilind Tambe

Maimuna Majumder

Working Paper 28135http://www.nber.org/papers/w28135

NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts Avenue

Cambridge, MA 02138November 2020

The views expressed herein are those of the authors and do not necessarily reflect the views of theNational Bureau of Economic Research.

NBER working papers are circulated for discussion and comment purposes. They have not been peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies officialNBER publications.

© 2020 by Andrew Perrault, Marie Charpignon, Jonathan Gruber, Milind Tambe, and Maimuna Majumder.All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicitpermission provided that full credit, including © notice, is given to the source.

Designing Efficient Contact Tracing Through Risk-Based QuarantiningAndrew Perrault, Marie Charpignon, Jonathan Gruber, Milind Tambe, and Maimuna Majumder NBER Working Paper No. 28135November 2020JEL No. I18

ABSTRACT

Contact tracing for COVID-19 is especially challenging because transmission often occurs in the absence of symptoms and because a purported 20% of cases cause 80% of infections, resulting in a small risk of infection for some contacts and a high risk for others. Here, we introduce risk-based quarantine, a system for contact tracing where each cluster (a group of individuals with a common source of exposure) is observed for symptoms when tracing begins, and clusters that do not display them are released from quarantine. We show that, under our assumptions, risk-based quarantine reduces the amount of quarantine time served by more than 30%, while achieving a reduction in transmission similar to standard contact tracing policies where all contacts are quarantined for two weeks. We compare our proposed risk-based quarantine approach against test-driven release policies, which fail to achieve a comparable level of transmission reduction due to the inability of tests to detect exposed people who are not yet infectious but will eventually become so. Additionally, test-based release policies are expensive, limiting their effectiveness in low-resource environments, whereas the costs imposed by risk-based quarantine are primarily in terms of labor and organization.

Andrew PerraultCenter for Research on Computation and SocietyHarvard UniversityCambridge, MA 02138aperrault@g.harvard.edu

Marie CharpignonInstitute for Data, Systems, and Society MITE18 – 407A50 Ames StreetCambridge, MA 02142United Statesmcharpig@mit.edu

Jonathan GruberDepartment of Economics, E52-434 MIT50 Memorial DriveCambridge, MA 02142and NBERgruberj@mit.edu

Milind TambeCenter for Research on Computation and Society33 Oxford StCambridge, MA 02138United Statesmilind_tambe@harvard.edu

Maimuna MajumderComputational Health Informatics ProgramBoston Children's Hospital300 Longwood AvenueLandmark 5506, Mail Stop BCH3187Boston, MA 02115maimuna.majumder@childrens.harvard.edu

A data appendix is available at http://www.nber.org/data-appendix/w28135 A Code is available at https://github.com/aperrault/test-trace

Designing Efficient Contact Tracing Through Risk-Based

Quarantining

Andrew Perrault, Marie Charpignon, Jonathan Gruber,

Milind Tambe, Maimuna S. Majumder

November 16, 2020

1 Introduction

As cases of coronavirus disease 2019 (COVID-19) pass 28 million worldwide (Dong, Du, and Gardner 2020), a clear

distinction has emerged between countries. There are those nations, such as South Korea, that have a very effective

regime structured around testing and contact tracing. Aggressive use of both has kept case counts low, minimized

damage, and crushed ongoing outbreaks as they emerge. Other nations, most notably the United States (U.S.), have

been much less organized in both their testing and contact tracing regimes. As a result, case counts have been high,

and transmission of SARS-CoV-2—the virus that causes COVID-19—has been frequent and ongoing.

Clearly, one of the major problems facing the U.S. is a lack of a coherent national testing strategy (Schneider 2020).

But another problem is a general failure to establish successful contact tracing, which has suffered from low response

rates and a large number of index cases that report having no contacts (Clark, Chiao, and Amirian 2020; Khazan

2020). Investigations regarding the determinants of quarantine adherence and contact reporting behavior suggest that

perceived effectiveness and cost of such behaviors are major drivers of participation (Smith et al. 2020; Webster et

al. 2020). SARS-CoV-2 faces challenges on both fronts. A typical non-household close contact has a seemingly low

probability (less than 6%) of becoming infected (Kucharski et al. 2020; Laxminarayan et al. 2020), most infections

are not serious, and because as much as 56% of transmission occurs without symptoms (Casey et al. 2020; Ferretti

et al. 2020), infectious individuals may not notice when they transmit the virus to others. Quarantining in the U.S.

is also expensive, especially given that a large number of workers do not receive full compensation for time missed

from work due to illness (U.S. Department of Labor and U.S. Bureau of Labor Statistics 2019). Combining these

factors yields a scenario where isolation (i.e., the separation of infected individuals from others) and quarantine (i.e.,

1

the separation of exposed individuals from others) times are perceived to be long and expensive relative to the benefit

they provide, resulting in low participation.

In the U.S., a monolithic approach to contact tracing may exacerbate these issues; individuals who are close

contacts of known infectious individuals are generally advised to quarantine irrespective of their odds of infection—

despite the fact that the probability of transmission varies widely for different contact events. Notably, studies suggest

that 80% of infections may be caused by only 10-20% of those who are infected with SARS-CoV-2 (Adam et al. 2020;

Endo et al. 2020; Laxminarayan et al. 2020), as is often the case with infectious diseases (Lloyd-Smith et al. 2005). As

a result, the effectiveness of quarantine may vary widely depending on the infected individual who is the source of the

exposure. To the best of our knowledge, transmission heterogeneity has not been exploited by previous contact tracing

systems, with the exception of Kennedy-Shaffer, Baym, and Hanage (2020), who rely on inexpensive, low sensitivity

tests to test all contacts for infection.

Of course, while some individuals have a much higher propensity to transmit (e.g., severely ill individuals in the

case of the related MERS-CoV (Majumder et al. 2017)), the underlying propensity of any given person is unknown.

Contract tracers can, however, observe a strong correlate of this propensity: the number of contacts of a given infected

person who develop symptoms. As this number increases, the risk of infection increases for any other individual in the

cluster. We argue for differential handling of contacts based on this observation, reserving more severe control mea-

sures, such as quarantine, for clusters where symptoms have been observed. We call our system risk-based quarantine

(RBQ).

In this paper, we evaluate the potential of RBQ using an agent-based branching process model of SARS-CoV-2

transmission (Kretzschmar et al. 2020; Okolie and Muller 2020; Klinkenberg, Fraser, and Heesterbeek 2006). We em-

bed contact tracing and quarantine into this model. We then introduce an RBQ regime, where clusters are observed for

symptoms for a day after being reached by contact tracers and the cluster is released from quarantine if no symptoms

are observed.1 Such an approach has two advantages; first, individuals spend less time in quarantine, which reduces

the costs of this approach, and second, individuals who are in a cluster where there is a reported infection may be more

likely to comply with quarantine because there is a clear signal that they are at risk. While our model was developed in

the context of the current SARS-CoV-2 epidemic, taking into account the transmission dynamics of the virus known

to date and testing processes currently in place, RBQ is also applicable to other infectious diseases with substantial

transmission heterogeneity and transmission in the absence of symptoms.

We compare our RBQ regime to test-driven approaches to reducing quarantine, such as releasing individuals from

quarantine who test negative. If tests were freely available, had no cost, and could predict with 100% accuracy whether

1. The ideal duration of observation may vary depending on the amount of time it takes to test the index case and reach their contacts and thedesired trade-off between transmission risk and amount of quarantine.

2

a contact has contracted or will develop SARS-CoV-2 infection, this would clearly be a better approach than requiring

individuals to quarantine; however, none of these three conditions can thus far be met in reality (Woloshin, Patel, and

Kesselheim 2020; Ramdas, Darzi, and Jain 2020; Wang et al. 2020; Ai et al. 2020; Service 2020; Waltz 2020). To

compare RBQ against more realistic testing-based policies, we incorporate testing, including false negatives and delays

in processing results, into our model and measure both the associated accuracy and cost. Much attention has been paid

to inexpensive, rapid, tests, which may alleviate test performance issues through volume of testing (Guglielmi 2020;

Larremore et al. 2020). Nevertheless, these tests are not yet available and implementing rapid testing at national scale

is a significant logistical challenge.

2 Modeling transmission and superspreading of SARS-CoV-2

To analyze the costs and effectiveness of various contact tracing policies, we use data from a number of U.S. and

non-U.S. sources (Table 1) to construct an agent-based branching process model of SARS-CoV-2 transmission over

three generations (Figure 1, detailed description in Appendix A). In each simulation, the index case (first generation)

represents an individual whose information has been provided to contact tracers. They are assumed to have age drawn

from the age distribution of the U.S. population (Howden and Meyer 2011), COVID-19 symptoms that led them to

isolate, and a positive test result. The other timing parameters of their infection and transmission are drawn from the

distributions specified in Table 1, including the onset of infectiousness, symptoms, and isolation. Age is used in the

model in two places: for the purposes of contact generation and to calculate the probability of death given infection.

To generate contacts, the index case is randomly assigned to a respondent from the POLYMOD contact sur-

vey (Mossong et al. 2008) with the same age category. This second generation consists of the contacts reported by that

respondent, repeated for each day of the index case’s pre-isolation infectious period. Each contact that was reported

as daily in POLYMOD is assumed to occur on every day of the pre-isolation infectious period, and contacts that were

reported as non-daily are assumed to represent unique individuals. We eliminate contacts with duration of less than 15

minutes, as they are not elicited by contact tracers under CDC guidelines (Centers for Disease Control and Prevention

2020).

Contact tracers have the opportunity to alter the behavior of second generation contacts in order to reduce trans-

mission to their own contacts (i.e., the third generation). We measure the effectiveness of a contact tracing policy

through the number of infections it prevents, relative to the amount of transmission that would occur in the absence of

control measures. The costs of a contact tracing policy include the days of quarantine served by the second generation

contacts and the monetary cost of performing tracing, monitoring and testing.

3

Index caseClose contacts Close contacts

of infectedclose contacts

InfectiousWithoutsymptoms

Time (days)

Withsymptoms

Exposed Quarantined In isolation

Served Droppedout

Figure 1: An agent-based branching process model is used to capture the interplay of stochastic behavioral dynamicsand contact tracing decision-making. The diagram depicts standard contact tracing with a two-week quarantine periodfor contacts for an example index case with six contacts during their three-day pre-isolation infectious period.

Figure 1 depicts the application of standard contact tracing with a quarantine period of two weeks from the time

of last exposure. The illustrative index case is infectious for two days without symptoms and one day with symptoms

before entering isolation. The index case has six contacts: two on day one of infectiousness, two on day two, one

on day three, and one on both day one and day two. Because processing the test results takes a day and contact

tracing takes a day (see Table 2 for our contact tracing modeling assumptions), the contacts are not asked to quarantine

until day 7. All contacts start their quarantine at the same time, but those who had a later last exposure are asked

to quarantine for longer. Contact #3 serves only a single day of quarantine before dropping out. Nevertheless, this

has no effect on transmission because they are not infectious. Contact #4 cannot be reached by tracers, but also never

becomes infectious. At the time of notification, Contact #5 has already become infectious and has exposed two people.

They develop symptoms the day after they are asked to quarantine and isolate for ten days after symptom onset. All of

the time they spends in quarantine is recorded by the simulation as isolation as we do not consider isolating infectious

individuals to have a social cost. The two individuals exposed by Contact #5 form the third generation of the model,

but neither becomes infected.

Modeling transmission heterogeneity. Studies on SARS-CoV-2 transmission suggest substantial transmission het-

erogeneity; 10–20% of cases are likely responsible for 80% of transmission (Adam et al. 2020; Endo et al. 2020;

Laxminarayan et al. 2020). To achieve a similar concentration of secondary infections, our model includes three

sources of transmission heterogeneity that correspond to hypothesized causes of superspreading events (Shen et

al. 2004; Frieden and Lee 2020). First, there is natural variation in the number of close contacts reported by each

respondent in POLYMOD, i.e., the top 20% of respondents report 50% of the close contacts. Second, household and

physically-close contacts are more likely to result in transmission. Third, each individual has a chance of being highly

transmissive, which represents a combination of behavioral and biological factors such as viral shedding and mask

4

wearing. We calculate the probability that a close contact results in transmission as the product of a base transmission

probability (which depends on whether the contact occurred within a household or not), a multiplier if the contact was

physical, and a multiplier if the transmitter was highly transmissive:

Pr(transmission|contact) = (phouseholdI[household] + pother(1� I[household]))MULTphysMULThighly transmissive (1)

Using Nelder-Mead (Nelder and Mead 1965), we calibrate the values of the parameters phousehold, pother, MULTphys,

MULThighly transmissive and Pr(highly transmissive) to match the calibration targets of an 18.8% secondary attack rate

among household close contacts (Madewell et al. 2020), a 6% attack rate among other close contacts (Kucharski

et al. 2020), and such that the top 20% of cases (i.e., the 20% that cause the most infections) cause 80% of infec-

tions (Adam et al. 2020).

The calibrated values are reported in Table 1. The resulting model matches the calibration targets exactly and has

an R0 of 1.88, somewhat lower than current estimates (Liu et al. 2020). This lower R0 is consistent with the fact that

most contact tracing studies observe a low Reff, which may be caused by interventions such as physical distancing or by

the difficulty in detecting all transmissions. Of all secondary infections, 17% are caused by asymptomatic cases (i.e.,

cases that never develop symptoms) and 37% by pre-symptomatic (i.e., cases that will eventually develop symptoms,

but have not at time of transmission), broadly consistent with the estimates of 10% and 46% by Ferretti et al. 2020. Of

the 20% of cases that result in the most secondary infections, 55% are highly transmissive; they also have 8.7% more

close contacts than average (26.3 vs. 24.2), 23% more household contacts (3.62 vs. 2.95), and 4.4% more physical

contacts (17.9 vs. 17.2).

There are potentially multiple calibrations matching the targets, because they do not sufficiently distinguish dif-

ferent causes of transmission heterogeneity, e.g., whether physical contact or individual variation in infectiousness

are more impactful. Additionally, we do not account for heterogeneous susceptibility across contacts, e.g., due to

immunocompromised status. In the discussion, we describe how our results depend on the cause of transmission

heterogeneity.

3 Using infection risk estimates to drive quarantine policies

A consequence of transmission heterogeneity is that the probability that an exposed individual becomes infected can

vary dramatically based on the source and the type of contact event. For example, in our model, a household contact of

a highly transmissive individual has nearly a 90% chance of becoming infected. Ideally, we would like contact tracing

policies to select different outcomes for contacts that depend on the probability of infection, e.g., enforce a strict

5

Table 1: Parameters of the infectiousness modelParameter Assumed value Details and references

Baseline individual-level dynamicsIncubation time Log-normal: Log mean 1.57

days and log std 0.65 daysBi et al. 2020. Mean: 5.94 days.

Symptom onset to isolation Gamma: shape 1.28 days andscale 1.22 days

Bi et al. 2020. Mean: 1.56 days.

Probability that a symptomatic infectioneventually self-isolates

0.9 We assume that symptomsurveillance programs, such astemperature checks, are ableto identify most symptomaticindividuals, and that mostcomply with self-isolation.

Duration of infectious period 7 days—2 days before and 5days after onset if symptomatic

Assumed. The length of the in-fectious period is not well un-derstood (Byrne et al. 2020).

Probability that a SARS-CoV-2 infectionis eventually symptomatic

0.8 Buitrago-Garcia et al. (2020).

Infectiousness of pre-symptomatic andasymptomatic SARS-CoV-2 infection,relative to symptomatic

0.63 (pre-symptomatic), 0.36(asymptomatic)

Buitrago-Garcia et al. (2020).

Infection fatality rate 0.01% for ages 0–24, 0.13% forages 25–54, 0.7% for ages 55–64, 6.8% for 65+

Levin et al. (2020).

Calibration targetsSecondary attack rate among householdcontacts

0.188 Madewell et al. (2020).

Secondary attack rate among other con-tacts

0.06 Kucharski et al. (2020).

Proportion of infections resulting fromtop 20% of cases

0.8 Adam et al. (2020).

Parameters of calibrated modelBase probability of transmission forhousehold contact

0.0356 This paper.

Base probability of transmission for othercontacts

0.00669 This paper.

Probability that an individual is highlytransmissive

0.109 This paper.

Infectiousness multiplier for highly trans-missive individuals

24.4 This paper.

Infectiousness multiplier for physicalcontact

5.21 This paper.

6

quarantine on contacts of highly transmissive individuals and monitor other contacts for symptoms via an app-based

check-in. Differential handling of contacts by risk level would, in principle, allow for achieving more transmission

reduction at a lower cost in total quarantine days.

Formally, let s be the source individual, e be the exposed individual, and c be the parameters of the close contact.

We can calculate the posterior probability that s infects e by the contact c:

Pr(e infected) = Pr(s infected)Pr(s infects e through c|s infected). (2)

However, we often lack information about the level of infectiousness of the source individual. Highly transmissive

individuals are usually only observed retrospectively because it is very difficult to proactively estimate an individ-

ual’s level of transmissiveness (Wong et al. 2015). As it appears that individual-level differences in transmissiveness

(both behavioral and biological) are a major driver of transmission heterogeneity, Equation 2 may not provide useful

information in practice.

We instead propose to retroactively identify highly transmissive individuals through the observed outcomes experi-

enced by close contacts. Intuitively, the longer we monitor the close contacts of a source individual without observing

symptoms, the less likely it is that the source individual is highly transmissive. Let HI(s) denote the event that s is

highly transmissive. Given symptom observations o for all close contacts, we can calculate the posterior probability

that s is highly transmissive using Bayes’ rule:

Pr(HI(s)|o) = Pr(o|HI(s))Pr(HI(s))

Pr(o|HI(s))Pr(HI(s)) + Pr(o|¬HI(s))Pr(¬HI(s). (3)

This estimate of the infectiousness level of the source can be inserted into Equation 2 to improve our estimate that e

is infected and thus drive a differential quarantine policy. We provide a numerical illustration of the combination of

Equations 2 and 3 at the end of this section.

Risk-based quarantine. Equations 2 and 3 can be used to design complex contact tracing policies that exploit all

of the available data and the entire space of decisions. We show that a simple policy we call “risk-based quarantine”

(RBQ) can yield contact tracing policies that are very effective at reducing transmission while requiring less quarantine

than standard quarantine-only policies. RBQ quarantines and observes the close contacts for n days. On day n + 1,

if no close contacts have developed symptoms, they are released into monitoring. The parameter n can be selected

according to local conditions or depend on cluster size, as larger clusters provide more independent observations of

infectiousness. We focus our attention on the case of n = 1, where contacts serve a single day of quarantine while

7

Household

Non-household

Day 1–3

Day 4

Day 5 Day 6

A B C

AD

E

Aobserves symptoms,

self-isolates,and is tested

A- receives positive

test results- contacts are

elicited

B C

DE

are reached,asked to

quarantine,and queried

about symptoms

Day 7

B C

DE

Because none of

display symptoms,they are releasedfrom quarantine

Figure 2: Risk-based quarantine can achieve a comparable reduction in transmission compared to quarantine-onlycontact tracing and requires fewer quarantine days. On day 1–3, Subject A is unknowingly infectious and exposesSubjects B–E. On day 4, A observes symptoms, self-isolates and is tested. On day 6, B–E are reached and checked forsymptoms. Because none of B–E display symptoms, the cluster is released on day 8.

they are observed for symptoms before the decision of whether to release the cluster is made.

Figure 2 shows an example of risk-based quarantine with n = 1 days of observation for a single index case, who is

labeled A in the figure. Subject A has two household contacts, B and C, and two non-household contacts, D and E,

on days 1–3. On days 1 and 2, A is infectious, but pre-symptomatic. On day 3, A develops symptoms, and on day 4, A

self-isolates and is tested (recall that we assume that all index cases self-isolate, seek testing, and remain isolated until

they are no longer infectious). On day 5, A receives a positive test result and their contacts are asked to quarantine

on day 6. In standard contact tracing, these contacts would serve two weeks of quarantine from the day of exposure.

Under RBQ, they are released after a day because none of B–E display symptoms. In the simplest form of RBQ,

contact tracers do not remain in contact with B–E after this point. In the next section, we describe more conservative

variants of RBQ that have different conditions for releasing asymptomatic clusters or that continue to monitor B–E

after the end of quarantine.

Because RBQ uses the symptoms of close contacts to drive decision-making, it is necessary that our model account

for the observation of symptoms that are not caused by SARS-CoV-2. These may be due to another illness such as

influenza or due to ”worried well” behavior. We set the chance that an individual experiences “false-positive” COVID-

19 symptoms to 1% per day, consistent with the higher end of estimates by Hinch et al. (2020). Because contact tracers

ask the close contacts if they have experienced any symptoms since the day of exposure, we observe false-positive

symptoms in about 10% of close contacts. Given that close contacts are observed on average for about a week, this

rate is somewhat higher than rates of reporting of influenza-like symptoms observed by Smolinski et al. (2015) of

13–17% per season.

Of course, the effectiveness of such an RBQ system depends on both the size of the cluster and how long it is

8

Figure 3: The probability that an individual becomes infected given that their entire cluster is asymptomatic by day x.Individuals from larger clusters are less likely to be infected when released from quarantine under RBQ.

observed for. A large cluster that does not display symptoms is a much stronger indication that the index case is not

highly transmissive than a small cluster that does not display symptoms. Likewise, the longer the period of observation

without symptoms in the cluster, the more likely an individual in the cluster is not infectious. To illustrate this point,

Figure 3 illustrates the probability that a released individual is actually infectious under RBQ. Each line represents a

different cluster size, while the lines show how results vary with duration of quarantine before release. For example,

if an index case has a single contact (top line of Figure 3), the probability that the contact will eventually be infected

given that they do not have symptoms on the day the contact tracer contacts them is about 20%. If we instead have a

cluster of size 9 (bottom line of Figure 3), the equivalent probability is 7.5%.

3.1 More conservative extensions of risk-based quarantine

Combining RBQ with other interventions can further reduce transmission at an additional cost, either monetary or in

quarantine days served.

Exit testing. By testing contacts who are eligible for release under baseline RBQ, some asymptomatic and pre-

symptomatic cases can be detected and isolated.

Observing smaller clusters for longer. A consequence of the simplicity of baseline RBQ is that the risk of infection

at the time of release varies dramatically by cluster size (Figure 3). Larger clusters provide more information about the

infectiousness of the index case because there are more samples; furthermore, smaller clusters have a larger share of

household close contacts on average. To reduce the variation in the amount of risk incurred, we can prescribe additional

9

observation days for clusters that are below a particular size. Observing smaller clusters for a few additional days can

further reduce transmission while only marginally increasing the amount of quarantine.

Active monitoring. Because COVID-19 symptoms are often initially very mild, symptoms can easily be missed by

infected individuals, which is reflected in the observed average delay of 1.5 days between onset and self-isolation (Bi

et al. 2020). We consider active monitoring, where each individual completes an app-based symptom screening each

day. Digital approaches to syndromic surveillance, such as Flu Near You (Smolinski et al. 2015), have proven to be

effective ways of eliciting long-term engagement with symptom surveillance systems through web or phone-based

polling (Lapointe-Shaw et al. 2020). We view the use of app-based technologies as a supplement to existing contact

tracing strategies, providing an additional communication platform to share resources and provide support.

We model active monitoring as reducing the isolation delay to one day for those for whom it exceeds one day.

Active monitoring incurs an additional cost, in that some individuals who display false-positive COVID-19 symptoms

are instructed to isolate. However, these “pseudo-isolation” days may have other beneficial effects, such as reducing

the spread of other respiratory infections.

3.2 Comparison to other contract tracing policies

Table 2 describes our contact tracing modeling assumptions. Our model of contact tracing is intended to represent a

robust, but imperfect, public health system. Because of ongoing physical distancing and masking efforts, we perform

a baseline reduction in non-household contacts relative to our calibrated model; namely, the probability that a given

contact event does not occur is 0.5, resulting in an average of 15.8 close contacts (SD 11.6) per individual. Reaching

80% of index cases and 80% of their contacts in one day corresponds to a highly effective, but not unrealistic, contact

tracing system. We assume that contact tracers are paid $20 USD per hour. Interviewing an index case takes 30

minutes plus 15 minutes per contact, and each contact that is in quarantine, isolation or active monitoring requires 10

minutes of contact tracer time per day.

A key aspect of modeling quarantine is capturing the dropout rate, which undercuts the effectiveness of the system.

We assume that the probability of dropping out of quarantine is 5% per day, although we also assume that individuals

who are symptomatic or who test positive do not drop out of isolation. We also assume that this dropout probability

is responsive to information about the infectiousness of the cluster; in particular, we assume that the probability of

dropout is halved when an individual is informed that another individual in their cluster has developed symptoms.

We discuss and perform sensitivity analysis on these assumptions in the discussion section, as they are not strongly

informed by data.

10

Table 2: Parameters of the intervention modelsParameter Assumed value Details and references

Probability that a contact event does notoccur

0.5 We assume a substantial re-duction in contacts relative tothe conditions of Mossong etal. (2008), but not as largeas the 75% reduction that wasreported at the peak of lock-down (Jarvis et al. 2020).

Contact tracingProbability that an index case is reachedand provides contacts

0.8We assume a reasonablyeffective contact tracinginfrastructure.

Probability that a contact is provided andcan be reached

0.8

Self-isolation to completed tracing 1 dayContact tracer time required to trace 30 minutes plus 15 minutes per

contactThe amount of time requiredfor interviews and check-insmay vary. Our estimates areintended to be conservative.Contact tracer time required for monitor-

ing, quarantine, and isolation10 minutes per individual perday

Contact tracer hourly wage $20 USD Average hourly wage from In-deed.com.

QuarantineQuarantine duration 14 days from last exposure Centers for Disease Control and

Prevention (2020).Isolation duration 10 days from symptom onset Centers for Disease Control and

Prevention (2020).Baseline dropout probability 5% per day Consistent with the observa-

tions of (Soud et al. 2009) dur-ing a mumps outbreak.

Symptomatic dropout probability 0% per dayAssumed. These quantities arepoorly understood and requirefurther study—see discussion.

Dropout probability after a negative test 10% per dayDropout probability after learning that an-other person who was exposed by thesame index case developed symptoms

2.5% per day

TestingRT-PCR test administration to results 1 day Time to process results

varies (Lazer et al. 2020).See Figure 4 for sensitivityanalysis.

RT-PCR test false negative rate 100% for 1 or more days be-fore onset of infectiousness, de-clining to 25% at 3–6 days af-ter onset of infectiousness, andincreasing 3.75% per day there-after

As in Figure 2 of Kucirka etal. (2020).

Test cost $200 USD Costs vary widely (Ruoff2020).

11

Table 3: Evaluated contact tracing policies

Contract tracing policy Description

No contact tracing No contact tracing or quarantine is performed.

Quarantine-only All close contacts are requested to quarantine until 14days have passed from the day of exposure.

Single-test release Close contacts are requested to quarantine and tested im-mediately. Individuals who RT-PCR test negative are re-leased.

Double-test release Close contacts are requested to quarantine and repeatedlytested during quarantine, starting immediately. Individu-als who RT-PCR test negative twice in a row are released.

Baseline RBQ As described in Section 3 with n = 1 days of observation.

RBQ + exit testing RBQ where contacts must additionally RT-PCR test neg-ative to be eligible for release.

RBQ + 4 extra observation daysfor clusters of size 8 or less

RBQ where clusters of size 8 or less are observed for 4extra days before deciding whether to release them.

RBQ + active monitoring RBQ where non-quarantined contacts are actively moni-tored. We consider a digital approach where each individ-ual completes a symptom screening each day. We assumethat there is a one-day delay between symptom onset andisolation for actively monitored contacts.

We assume that, at baseline, RT-PCR tests take one day to administer. As the number varies widely in prac-

tice (Lazer et al. 2020), we perform sensitivity analysis on this quantity. Our model of RT-PCR test is based on the

analysis of Kucirka et al. 2020 and presents a high false-negative rate initially that drops to 25% at 3–6 days after

symptom onset.

We compare baseline RBQ to several alternative contact tracing policies (Table 3). We include a baseline without

tracing at all and standard quarantine-only tracing, which provide the two extremes of our analysis. Additionally, we

include test-based release policies. Single-test release tests individuals immediately when they are reached by tracers

and releases individuals who test negative. Double-test is a more conservative variant; after the results of each test

become available, another test is taken, until two negative tests are obtained or quarantine ends. We additionally con-

sider several variants of RBQ, including the conservative extensions from the previous section. For longer observation

times for smaller clusters, we choose four extra observation days for clusters of size eight or less as a compromise

between transmission reduction and additional quarantine.

For each policy, we simulate 100,000 index cases. The model code is available online (https://github.

com/aperrault/test-trace).

12

Table 4: Comparing the effectiveness of contract tracing policies

Contract tracing policy Reff ofclosecontacts

Meanreductionin Reff

Quarantinedays perindex case

Deaths per1000 indexcases

Cost per in-dex case

No contact tracing 1.36 0% 0 27.4 $0Quarantine-only 0.926 31.8% 62.1 22.6 $189RBQ 1 26.1% 36.1 23.8 $144RBQ + exit testing 0.967 28.8% 40.1 23.2 $957RBQ + 4 extra observation daysfor clusters of size 8 or less

0.968 28.7% 40.5 23.2 $152

RBQ + active monitoring 0.967 28.8% 36.1 23.2 $208RBQ + exit testing + 4 extraobservation days for clusters ofsize 8 or less + active monitor-ing

0.926 31.8% 42.6 22.5 $970

Single-test release 1.17 13.8% 14.9 25.8 $1630Double-test release 1.1 19.1% 21.2 24.8 $3500

4 Results

Our model predicts that RBQ reduces transmission more than test-based policies and costs many fewer quarantine

days than standard quarantine-only tracing. RBQ results in a slightly higher level of transmission and deaths than

quarantine-only tracing, but this gap can be bridged by combining the more conservative extensions to RBQ, incurring

a marginal additional requirement of quarantine days and monetary cost.

Table 4 presents a comparison of alternative contact tracing and testing policies. We show the Reff of second

generation contacts, as well as the reduction in Reff relative to a baseline where no contact tracing is present. We then

show the total days of quarantine per index case among all second generation contacts. We compute deaths per 1000

index cases across second and third generation contacts. Finally, we show the cost per index case, including costs of

tracing, testing and monitoring.

We find that quarantine-only contact tracing reduces transmission among second generation contacts by 31.8% on

average, resulting in 17.5% fewer deaths per index case among second and third generation contacts. The cost of this

reduction in transmission and deaths is 62.1 quarantine days and $189, which is spent entirely on contact tracers.

In contrast, RBQ results in a slightly higher rate of deaths, but a significant reduction of days in quarantine. In

particular, baseline RBQ results in a somewhat higher Reff of 1 (vs. 0.926 for quarantine-only) and 5% more deaths,

but only requires 36.1 days of quarantine, representing a 41.9% reduction. Baseline RBQ is somewhat less expensive

(24%) than quarantine-only because less contact tracer time is required to communicate with quarantined and isolated

individuals.

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The effectiveness of RBQ in terms of reducing deaths can be enhanced in a variety of ways. Adding exit testing,

additional observation days to small clusters, or app-based active monitoring to RBQ yields similar reductions in

Reff (28.7–28.8%) and deaths per index case (23.2 per thousand, or 2.6% less than baseline RBQ). The cost of these

additions varies. Exit testing has a high monetary cost of $957 per index case and adds 11% more quarantine days. The

additional quarantine days are required because each individual who is eligible for release under RBQ must remain

quarantined until they receive their test results. The additional observation days are inexpensive at $152 per index

case and add 12% more quarantine days, slightly more than exit testing. Active app-based monitoring does not cause

additional quarantine days relative to baseline RBQ and costs 44.4% more ($208 per index case). Active monitoring

adds 1.6 days of monitoring-induced isolation per index case, consisting of isolation time spent by individuals who

are not infected with SARS-CoV-2.

Combining all three extensions to baseline RBQ results in the same reduction in transmission as quarantine-only

(31.8%) and fewer deaths (22.5 per 1000 index cases) but costs 31% fewer quarantine days (42.6 vs. 62.1). The

monetary cost of the combined approach is $970 per index case. The combined approach is able to achieve the same

transmission reduction with fewer quarantine days because of our assumption that quarantine adherence is improved

by the RBQ protocol 2

We find that contact tracing policies that immediately test contacts are less effective than RBQ at preventing

transmission. Single-test release, which immediately tests each traced contact and releases those who test negative,

prevents just 13.8% of transmission relative to the baseline without contact tracing—less than half of the transmission

reduction achieved by quarantine-only. The 5.8% reduction in deaths (to 25.8 per 1000 index cases) is similarly

small. Furthermore, single-test release is 8.6 times more expensive than quarantine-only at $1630 per index case. The

advantage of single-test release is the small number of quarantine days per index case. Each uninfected contact serves

a single day of quarantine after they are tested, resulting in 14.9 quarantine days per index case.3 The initially high

false positive rate of RT-PCR tests combined with the variation in time to onset of infectiousness makes it difficult to

use test results to determine whether an individual should be released from quarantine.

Double-test release, where contacts who test negative twice consecutively are released from quarantine, is some-

what more effective at reducing transmission, but is much more expensive. It achieves a reduction in Reff of 19.1%

and a reduction in deaths of 9%. However, the cost is more than twice that of single-test release at $3500 per index

case. Double-test release costs about 50% more quarantine days (21.2 per index case) than single-test release.

Test-based policies require rapid test turnaround to be competitive with RBQ in terms of number of quarantine

2. Recall that after stating that symptoms were observed in a cluster, we assume that quarantine dropout rate is halved from 5% per day to 2.5%per day.

3. Note that we do not consider introducing a delay before the first test is conducted. Doing so would reduce transmission and increase quarantine.

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Figure 4: Delays in test processing reduces the effectiveness of all contact tracing policies and rapidly increases thenumber of quarantine days induced by test-based policies.

days (Figure 4). Recall that our baseline scenario assumes an optimistic 24-hour time between test and results. This

point is illustrated in Figure 4. Each panel in the figure shows the effect of lengthening time between test and results,

first on Reff, and then on average quarantine days. Recall that our baseline scenario assumes an optimistic 24-hour

time between test and results, but in practice times have typically been much longer. At a test turnaround of three days,

double-test release performs similarly to RBQ, with Reff of 1.09 (vs. 1.10), 24.7 (vs. 24.9) deaths per 1000, and 39.2

(vs. 36.3) quarantine days per index case. Double-test release is 19.2 times more expensive, however.

We find that test processing delays causes quarantine-only and RBQ to become less effective due to the need to

receive a positive test result before tracing the contacts of the index case.4 Test-based policies initially see an increase

in effectiveness as a result of test processing delays. Because it takes longer to receive a positive test result for the index

case, the tests of second generation contacts are performed later, yielding results that more accurately reflect whether

these individuals are infectious. The number of quarantine days for test-based release policies initially increases as

individuals spend more time quarantined, waiting for test results, and then decreases because the exposure date was

sufficiently long ago that quarantine ends before test results are received.

5 Discussion

Using a combination of data sources, we construct a branching process model of SARS-CoV-2 transmission and

contact tracing. Our results are striking. We find that at baseline, with no contact tracing, each infected person causes

0.0274 deaths among their contacts, and each contact causes 1.36 infections on average. With contact tracing that

4. It is possible to consider contact tracing that does not require a positive test result for the index case, e.g., tracing individuals that have a highlypredictive set of symptoms and an attribute that makes them more likely to be highly transmissive.

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requires a full 14-day quarantine, the death rate falls by 18% to 0.0226, and transmission reduces by 31.8% to 0.926

infections per contact—a substantial reduction—but the set of contacts spends an average of 62.1 days in quarantine.

The long quarantine time may be quite costly and disincentivize both index cases and contacts from participating in

contact tracing. On the other hand, with a baseline RBQ system, each infected person causes 5% more deaths and each

contact causes 8% more transmission, but contacts spend only 36.1 days in quarantine, resulting in a 42% reduction.

At a cost of 18% more quarantine days and $826 more per index case, a more conservative version of RBQ can achieve

the same number of infections and deaths as standard contact tracing.

An alternative approach to reducing the number of quarantine days is to use testing to identify infected contacts and

release those contacts who test negative. However, requiring a single negative test, conducted immediately, is much

less effective at reducing transmission and deaths than the alternatives we consider. Transmission is reduced by 13.8%

and deaths by 6%—less than half the reductions achieved by even a baseline RBQ system. Requiring two consecutive

negative tests to release performs somewhat better, achieving a 19.1% reduction in transmission and a 10% reduction

in deaths. While these test-based policies are effective at reducing the number of quarantine days served, with a 76%

reduction for single-test release and a 65% reduction for double-test release, this advantage is quickly eliminated if

there are delays in test processing. Furthermore, test-based policies are much more expensive. The combination of cost

and sensitivity to test delays makes test-based policies challenging to implement in under-resourced environments.

RBQ could be extended by using properties of the contact events to inform decision-making. The more detailed the

information that is available about exposure events and the more accurate a predictive model of symptoms and infection

that can be constructed, the more effectively an RBQ system can perform. Despite the detail in our transmission

model, we assume that contact tracers do not integrate information about the context of the contact (e.g., household

vs. non-household) in their evaluation of the risk associated with the index case. By making this assumption, our

findings are more robust to errors in our model of transmission. Nevertheless, in a setting where more precise models

of transmission can be constructed and are available to tracers, RBQ can potentially be made considerably more

effective.

Our modeling approach has significant limitations. We highlight four. First, we aggregate data from heterogeneous

and possibly incompatible sources with respect to the onset and presentation of symptoms; further, there are some areas

where very little relevant data are available. Our estimates of the effectiveness associated with RBQ depend critically

on the fraction of individuals who develop COVID-19 symptoms after exposure and the timing of when they do so.

To evaluate the effectiveness of RBQ, we need to know the probability of COVID-19 given observed symptoms and

known exposure to an individual who tested positive for SARS-CoV-2 infection. The less predictive COVID-19-like

symptoms are of infection, the less effective RBQ will be at reducing the amount of quarantine served. It may be

16

desirable to observe contacts for a specific set of highly predictive symptoms, such as anosmia (Petersen and Phillips

2020). Doing so would yield higher confidence that a symptomatic individual is infected at the cost that more infected

individuals would be asymptomatic under this definition.

Second, we are missing important data on the subject of adherence to quarantine and isolation. Ideally, our model

would be parameterized using a distribution of the number of contacts of quarantined and isolated individuals as a

function of quarantine length and motivation, i.e., information provided to the individual about the likelihood that

they are sick or infectious. Unfortunately, most existing studies of quarantine adherence ask all-or-nothing questions,

such as whether all conditions of quarantine were observed over the entire period, which do not inform us about the

severity or timing of quarantine violations. To address this data insufficiency, we perform a sensitivity analysis around

quarantine adherence assumptions in Appendix B. We find that RBQ performs better relative to quarantine-only when

dropout rates are higher. This can be explained by the ability of RBQ to better exploit the lower dropout rate that

we assume is a result of an individual believing they are more likely to be infected. If an individual’s belief that

they are infected does not affect their quarantine dropout rate, RBQ becomes less effective at reducing transmission.

Nevertheless, increases in participation in contact tracing due to shorter quarantines may still result in less transmission

at the population level.

Third, the dynamics of COVID-19 transmission heterogeneity are still not well understood. The model attributes

transmission heterogeneity to three causes. First, an individual can have more contacts. Second, they can have contacts

that are more intense on average, e.g., because they are physical. Third, they can be more transmissive, either due to

biological or behavioral factors. Our model includes all three causes, but we do not have strong evidence to support

attribution of heterogeneity to each. A fourth cause that is not included in our model is variation in susceptibility, e.g.,

among the elderly or immunosuppressed. For RBQ, the specific balance between second, third and fourth causes does

not affect our results. All of these lead to variation in the propensity of an index case to transmit the infection to their

contacts, which is what RBQ exploits. Nevertheless, it is possible that transmission heterogeneity can be explained

by variation in the number of contacts only. For example, if SARS-CoV-2 transmission is primarily inhalational, the

definition of contact that is used by contact surveys, which requires prolonged physical proximity, may be inappro-

priate. This appears unlikely as studies of superspreading events to date have generally demonstrated prolonged close

contact (Frieden and Lee 2020).

Fourth, our transmission model lacks detail in some areas. We assume a constant rate of asymptomatic infections

across age. Age-specific asymptomatic rates would increase the effectiveness of RBQ in populations that are more

likely to show symptoms and decrease it in those that are less likely. Similarly, we assume a binary, rather than

continuous, model of high infectiousness that does not vary with time. The effectiveness of RBQ does not depend on

17

this assumption as long as transmission heterogeneity remains at levels that are similar or above those we consider.

Significant obstacles must be overcome to implement RBQ in practice. System inertia is a major factor—there is

a body of contact tracing best practices that do not consider differential handling based on risk. In order to understand

whether the benefits are worth the transitioning costs, we recommend a two-pronged approach. First, the effectiveness

of RBQ at preventing transmission can be tested empirically. In an information-rich environment, e.g., a school or

university, where app-based monitoring is already in use, RBQ can be tested empirically against standard quarantine-

only tracing by comparing the number of contacts of contacts who develop COVID-19 symptoms as reported by

the app. Because these environments are often coercive, they are not ideal for measuring the extent to which RBQ

may increase tracing participation. Thus, we secondly recommend using a survey to evaluate the extent to which

RBQ might impact participation in contact tracing. We would like to measure to what extent a substantial reduction

in quarantine days on average would influence the decision of individuals to provide information to contact tracing

systems.

In summary, our findings suggest that approaches to contact tracing that take infection risk into account, such

as RBQ, can reduce the amount of quarantine time served without increasing transmission and can do so inexpen-

sively. By reducing the cost that quarantine imposes on individuals, they may increase participation in contact tracing,

enabling less frequent and severe lockdown measures.

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